sympy-01.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "fda59f8f",
   "metadata": {},
   "source": [
    "# sympy"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5977c514",
   "metadata": {},
   "source": [
    "pip install sympy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "beb0cdbd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.8284271247461903"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import math\n",
    "math.sqrt(8)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "5fc4f757",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2 \\sqrt{2}$"
      ],
      "text/plain": [
       "2*sqrt(2)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import sympy\n",
    "sympy.sqrt(8)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "a991c703",
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy as sp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "fed72f86",
   "metadata": {},
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'x' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "Input \u001b[1;32mIn [5]\u001b[0m, in \u001b[0;36m<cell line: 1>\u001b[1;34m()\u001b[0m\n\u001b[1;32m----> 1\u001b[0m \u001b[43mx\u001b[49m\u001b[38;5;241m+\u001b[39m\u001b[38;5;241m1\u001b[39m\n",
      "\u001b[1;31mNameError\u001b[0m: name 'x' is not defined"
     ]
    }
   ],
   "source": [
    "x+1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "2d264f54",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x + 1$"
      ],
      "text/plain": [
       "x + 1"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = sp.symbols('x')\n",
    "x+1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "241a2f81",
   "metadata": {},
   "outputs": [],
   "source": [
    "x,y,z = sp.symbols('x y z')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "ea02bdfa",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x + y + z$"
      ],
      "text/plain": [
       "x + y + z"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x+y+z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "a7cfc1e4",
   "metadata": {},
   "outputs": [],
   "source": [
    "a,b =sp.symbols('b a')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "0b89f493",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle b$"
      ],
      "text/plain": [
       "b"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "de7e8141",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a$"
      ],
      "text/plain": [
       "a"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "13e90497",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1/2"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "687181f9",
   "metadata": {},
   "source": [
    "### اعداد کسری"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "8c660944",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{1}{2}$"
      ],
      "text/plain": [
       "1/2"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.Rational(1,2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "4d98a41b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{7}{6}$"
      ],
      "text/plain": [
       "7/6"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.Rational(1,2)+sp.Rational(2,3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "dfdc96a5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x + \\frac{1}{2}$"
      ],
      "text/plain": [
       "x + 1/2"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x+sp.Rational(1,2)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fbb9fb0d",
   "metadata": {},
   "source": [
    "### جایگذاری"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "116c53a4",
   "metadata": {},
   "outputs": [],
   "source": [
    "x, y =sp.symbols('x y')\n",
    "expr = sp.cos(x)+1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "9c36b7cd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\cos{\\left(x \\right)} + 1$"
      ],
      "text/plain": [
       "cos(x) + 1"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "15ae93f3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2$"
      ],
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.subs(x,0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "7ba0f750",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\cos{\\left(y \\right)} + 1$"
      ],
      "text/plain": [
       "cos(y) + 1"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.subs(x,y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "7a71c79a",
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = sp.sin(2*x) + sp.cos(2*x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "75171ea2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\sin{\\left(2 x \\right)} + \\cos{\\left(2 x \\right)}$"
      ],
      "text/plain": [
       "sin(2*x) + cos(2*x)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "fc6f18d7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2 \\sin{\\left(x \\right)} \\cos{\\left(x \\right)} + 2 \\cos^{2}{\\left(x \\right)} - 1$"
      ],
      "text/plain": [
       "2*sin(x)*cos(x) + 2*cos(x)**2 - 1"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.expand_trig(expr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "ebd2746b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2 \\sin{\\left(x \\right)} \\cos{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$"
      ],
      "text/plain": [
       "2*sin(x)*cos(x) + cos(2*x)"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.subs(sp.sin(2*x),2*sp.sin(x)*sp.cos(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "ddcae615",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{3} + 4 x y - z$"
      ],
      "text/plain": [
       "x**3 + 4*x*y - z"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x,y,z = sp.symbols('x y z')\n",
    "expr = x**3 + 4*x*y -z\n",
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "f82cca79",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 8 y - z + 8$"
      ],
      "text/plain": [
       "8*y - z + 8"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.subs(x,2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "c8080e03",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 40 - z$"
      ],
      "text/plain": [
       "40 - z"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.subs(x,2).subs(y,4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "ee62d1d7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 40$"
      ],
      "text/plain": [
       "40"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.subs(x,2).subs(y,4).subs(z,0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "185c64ea",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 40$"
      ],
      "text/plain": [
       "40"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.subs({x:2,y:4,z:0})"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8801cb34",
   "metadata": {},
   "source": [
    "### sympify"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "29f4f629",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'x**2+3*x-1/2'"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr_str = 'x**2+3*x-1/2'\n",
    "expr_str"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "70ad9155",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{2} + 3 x - \\frac{1}{2}$"
      ],
      "text/plain": [
       "x**2 + 3*x - 1/2"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = sp.sympify(expr_str)\n",
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "ad106cd1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{19}{2}$"
      ],
      "text/plain": [
       "19/2"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.subs(x,2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "id": "b9b44a01",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Enter expression:x+2\n"
     ]
    }
   ],
   "source": [
    "expr_str = input(\"Enter expression:\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "id": "d9f99555",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'x+2'"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr_str"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "id": "788b7961",
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = sp.sympify(expr_str)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "id": "a0ec5c0d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 4$"
      ],
      "text/plain": [
       "4"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.subs(x,2)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "22f8f108",
   "metadata": {},
   "source": [
    "### evalf"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "id": "38579c8f",
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = sp.sqrt(8)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "e30422da",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2 \\sqrt{2}$"
      ],
      "text/plain": [
       "2*sqrt(2)"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "id": "adc5a70f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2.82842712474619$"
      ],
      "text/plain": [
       "2.82842712474619"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.evalf()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "id": "23d863e5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145$"
      ],
      "text/plain": [
       "2.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.evalf(100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "id": "c109a910",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\cos{\\left(2 x \\right)}$"
      ],
      "text/plain": [
       "cos(2*x)"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = sp.cos(2*x)\n",
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "id": "6ca8eb19",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\cos{\\left(2 x \\right)}$"
      ],
      "text/plain": [
       "cos(2*x)"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.evalf()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "id": "009584fd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 0.08749898343944639236583365024741093189411162966538926084323152941193796669093594556950578905611342514$"
      ],
      "text/plain": [
       "0.08749898343944639236583365024741093189411162966538926084323152941193796669093594556950578905611342514"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.evalf(100,subs={x:2.4})"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1894d9c1",
   "metadata": {},
   "source": [
    "### lambdify"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "id": "90cf1659",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "id": "4c82ef61",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = np.arange(10)\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "id": "9ab4e9d6",
   "metadata": {},
   "outputs": [],
   "source": [
    "x = sp.symbols('x')\n",
    "expr = sp.sin(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "id": "e1b00eec",
   "metadata": {},
   "outputs": [],
   "source": [
    "f = sp.lambdify(x,expr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "id": "16544b28",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.        ,  0.84147098,  0.90929743,  0.14112001, -0.7568025 ,\n",
       "       -0.95892427, -0.2794155 ,  0.6569866 ,  0.98935825,  0.41211849])"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "id": "9e258fda",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.7568024953079282"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(4)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "567cee81",
   "metadata": {},
   "source": [
    "### ساده سازی"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "id": "5feb25ea",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\sin^{2}{\\left(x \\right)} + \\cos^{2}{\\left(x \\right)}$"
      ],
      "text/plain": [
       "sin(x)**2 + cos(x)**2"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = sp.sin(x)**2 + sp.cos(x)**2\n",
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "id": "8bb6bd6f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 1$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.simplify(expr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "id": "02fca4bd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{x^{3} + x^{2} - x - 1}{x^{2} + 2 x + 1}$"
      ],
      "text/plain": [
       "(x**3 + x**2 - x - 1)/(x**2 + 2*x + 1)"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = (x**3+x**2-x-1)/(x**2+2*x+1)\n",
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "id": "e9275cc8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x - 1$"
      ],
      "text/plain": [
       "x - 1"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.simplify(expr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "id": "54961cb5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{2} + 2 x + 1$"
      ],
      "text/plain": [
       "x**2 + 2*x + 1"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.simplify(x**2+2*x+1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d298a128",
   "metadata": {},
   "source": [
    "### factor"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "id": "eba65c85",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(x + 1\\right)^{2}$"
      ],
      "text/plain": [
       "(x + 1)**2"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.factor(x**2+2*x+1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8a9eb37b",
   "metadata": {},
   "source": [
    "### expand"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "id": "688cf528",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{2} + 2 x + 1$"
      ],
      "text/plain": [
       "x**2 + 2*x + 1"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.expand((x+1)**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "id": "35c53d4b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 1$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.simplify((x**2+2*x+1)/(x+1)**2)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9d82ae19",
   "metadata": {},
   "source": [
    "### collect"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "id": "d77611a4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{3} - x^{2} z + 2 x^{2} + x y + x - 3$"
      ],
      "text/plain": [
       "x**3 - x**2*z + 2*x**2 + x*y + x - 3"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = x*y+x-3+2*x**2-z*x**2+x**3\n",
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "id": "4e8c5c2b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{3} + x^{2} \\cdot \\left(2 - z\\right) + x \\left(y + 1\\right) - 3$"
      ],
      "text/plain": [
       "x**3 + x**2*(2 - z) + x*(y + 1) - 3"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.collect(expr,x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "id": "d0f68f6b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{3} - x^{2} z + 2 x^{2} + x y + x - 3$"
      ],
      "text/plain": [
       "x**3 - x**2*z + 2*x**2 + x*y + x - 3"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.collect(expr,z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "id": "c289f3da",
   "metadata": {},
   "outputs": [],
   "source": [
    "collected_expr = sp.collect(expr,x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "id": "d0abeacc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2 - z$"
      ],
      "text/plain": [
       "2 - z"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "collected_expr.coeff(x,2)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fdef1e07",
   "metadata": {},
   "source": [
    "### cancel"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "id": "a3a8cb17",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{x^{2} + 2 x + 1}{x^{2} + x}$"
      ],
      "text/plain": [
       "(x**2 + 2*x + 1)/(x**2 + x)"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = (x**2+2*x+1)/(x**2+x)\n",
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "id": "e5da1e53",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{x + 1}{x}$"
      ],
      "text/plain": [
       "(x + 1)/x"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.cancel(expr)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0e787801",
   "metadata": {},
   "source": [
    "### apart"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "id": "1688ce87",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{4 x^{3} + 21 x^{2} + 10 x + 12}{x^{4} + 5 x^{3} + 5 x^{2} + 4 x}$"
      ],
      "text/plain": [
       "(4*x**3 + 21*x**2 + 10*x + 12)/(x**4 + 5*x**3 + 5*x**2 + 4*x)"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = (4*x**3+21*x**2+10*x+12)/(x**4+5*x**3+5*x**2+4*x)\n",
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "id": "a88788c2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{2 x - 1}{x^{2} + x + 1} - \\frac{1}{x + 4} + \\frac{3}{x}$"
      ],
      "text/plain": [
       "(2*x - 1)/(x**2 + x + 1) - 1/(x + 4) + 3/x"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.apart(expr)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "584db9c9",
   "metadata": {},
   "source": [
    "### ساده سازی مثلثاتی"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "id": "ba52102b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x$"
      ],
      "text/plain": [
       "x"
      ]
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.cos(sp.acos(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "id": "c7da9500",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\pi}{2}$"
      ],
      "text/plain": [
       "pi/2"
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.asin(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "id": "fe62caae",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\sin^{4}{\\left(x \\right)} - 2 \\sin^{2}{\\left(x \\right)} \\cos^{2}{\\left(x \\right)} + \\cos^{4}{\\left(x \\right)}$"
      ],
      "text/plain": [
       "sin(x)**4 - 2*sin(x)**2*cos(x)**2 + cos(x)**4"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = sp.sin(x)**4 -2*sp.cos(x)**2*sp.sin(x)**2+sp.cos(x)**4\n",
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "id": "fd5a4dca",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\cos{\\left(4 x \\right)}}{2} + \\frac{1}{2}$"
      ],
      "text/plain": [
       "cos(4*x)/2 + 1/2"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.trigsimp(expr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "e927e3ea",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.9.12"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
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